nLab formal neighbourhood of the diagonal

Idea

Given any object $X$ in a cartesian monoidal category where formal completion is defined, then the formal neighbourhood of the diagonal of $X$ is the formal completion of the diagonal map $\Delta_X \colon X \longrightarrow X \times X$.

Hence intuitively, the formal neighbourhood of the diagonal of $X$ is the space whose points are pairs of infinitesimally close points in $X$.

Specifically for $X$ a scheme, then the formal neighbourhood of its diagonal is the formal scheme around $\Delta_X$. In this case the coequalizer of the two projections out of the formal neighbourhood is called the de Rham stack of $X$. Moreover, the (sheaf of modules of) sections of the formal neighbourhood, with respect to one of the two projection maps, is the tangent complex of $X$. This holds indeed more generally, at least also for $X$ a derived algebraic stack (Hennion 13).

More generally, see at infinitesimal disk bundle.

Related concepts

• infinitesimal object

• infinitesimal singular simplicial complex

• infinitesimal neighborhood

References

• Benjamin Hennion, Tangent Lie algebra of derived Artin stacks (arXiv:1312.3167)